Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×C9⋊C8 |
| derived | C9 ⋊ C23×C3×C8 |
| Sylow | C3×C9 ⋊ C23×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C4 | D9 |
| commutator, hypocenter | C9 | C23×C3×C8 |
| Frattini | C3×C4 | C23×C3×D3 |
| Fitting | C23×C3×C4×C9 | C2 |
| socle, prime radical | C24×C32 | Dic3 |
| cosocle | C4×C9 | C24×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C23×C3×C4 | |
1250 conjugacy classes of subgroups; only the 810 normal subgroups are shown.
This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 810 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
576 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19586);